1999 Fiscal Year Final Research Report Summary
Self maps of the suspension of H-spaces
| Project/Area Number |
10640077
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Geometry
|
|---|
| Research Institution | Wakayama University |
|---|
Principal Investigator |
MORISUGI Kaoru Faculty of Education, WAKAYAMA UNIVERSITY, Professor, 教育学部, 教授 (00031807)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
KAWAKAMI Tomohiro Faculty of Education, WAKAYAMA UNIVERSITY, Associate Professor, 教育学部, 助教授 (20234023)
OSHIMA Hideaki Ibaraki University, Faculty of Science, Professor, 理学部, 教授 (70047372)
HEMMI Yutaka Kochi University, Faculty of Science, Professor, 理学部, 教授 (70181477)
|
|---|
| Project Period (FY) |
1998 – 1999
|
| Keywords | H-spaces / Samelson products / Whitehead products / fiber bundles / homotopy groups / self maps |
|---|
| Research Abstract |
The summary of research results is as follows.
(a)Let X be an Hopf space. Oshima showed that the homotopy set of self maps of X [X, X] forms a group when X has at most 3-cells and he also determined the group structure of [X, X] for such spaces X.
(b)We studied the group structure [ΣX, ΣX], where ΣX is the suspension space of X. We determined the group structure in case that X = SU(3) and Sp(2). They are typical example of above space X. In general, the suspension map [X, X] → [ΣX, ΣX] is not a homomorphism.
(c)For general Hopf spaces, we can consider the Samelson products in [Y, X] when Y is a CW-complexes. The we found a formula which relates the Samelson product in [Y, X] and its suspension in [Y, ΩΣX] by using the generalized Hopf construction. The above item is an application of this formula.
(d)It is important to know what nilpotency the group [X, X] for a Hopf space X. However another important is to study the composition structure of the group [X, X]. We determined this composition structure for X = SU(3) and Sp(2). This structure looks like "Square ring" which Baues in Germany studied.
(e)Let MィイD1nィエD1 = SィイD1n-1ィエD1 UィイD22ィエD2eィイD1nィエD1 be the mod 2 Moore space. There is a canonical projection map P : MィイD1nィエD1 → SィイD1nィエD1. Given an element α∈ πィイD2κィエD2(SィイD1nィエD1), we studied when α has a lift to MィイD1nィエD1. One of our results is that the Whitehead square [ιィイD2nィエD2, ιィイD2nィエD2] does not have a lift for n ≠1, 3, or 7. This is the joint work with J.Mukai in Shinshu University.
|
